Proper families and almost additive functions (Q1349037)

The following definitions are used: Let \(S\) be a semigroup. \(S\) is named left reversible if \((a+S)\cap (b+S)\neq\varphi\), \(a,b\in S\). A family \(\lambda\) of subsets of \(S\) is left translation invariant if \(-a+U= \{u\mid a+u\in U\}\), for \(a\in S\), \(U\in\lambda\). A family \(\lambda\) in \(S\) is named \(k\)-proper if for every \(A_1,\dots, A_k\in\lambda\): \(A_1\cup \cdots\cup A_k \neq S\). A given condition holds for \(\lambda\)-almost all \((\lambda\)-a.a) \(x \in S\) if there exists \(U\in\lambda\) such that this condition holds for \(x\in S\setminus U\). For a set \(Q\subset S\times S\) and a family \(\lambda\) in \(X\) the sets \(Q[x]=\{y\in S\mid(x,y)\in Q\}\), \(\Omega (\lambda)= \{Q\subset S\times S\mid Q[x] \in\lambda\) for \(\lambda\)-a.a \(x\in S\}\), are considered. The main result of this paper is the following: If \(f:S\to H\) satisfies \(f(x+y)= f(x)+f(y)\) for \(\Omega(\lambda)\)-a.a \((x,y)\in S\times S\), where \(S\) is a left reversible semigroup, \(H\) is a group and \(\lambda\) is a left translation invariant family in \(S\) which is 4-proper, then there exists a unique additive function \(F:S\to H\), such that \(f(x)=F(x)\) for \(\lambda\)-a.a \(x \in S\).